an example of two-dimensional consolidation testing and

Paper

An Example of Two-dimensional Consolidation

Testing and Numerical Procedures

for Embankment Materials

Sohji INOUE*

Summary The technique for measuring the over-all value of the

coefficient of consolidation both in the horizontal and vertical direction has

attracted a great deal of attention. Accordingly, two-dimensional consolida-

tion tests and a numerical calculation method of the test results have been

developed to determine the horizontal and vertical coefficients of consolidation,

ch and cv, respectively, of transversely isotropic specimens. For determining

the condition of two directions of drainage, an oedometer ring with longitudinal

slits cut on the ring's wall, 0.15•`0.20cm wide and 10.0cm high, is used. The

vertical coefficient of consolidation, cv is determined from the one-dimensional

conventional consolidation testing procedure which is based on the JIS

standard.

The results of tests and calculations show that the values of ch are about 30

times greater than those of cv on average, ranging from 15 to 50 times as great.

The horizontal and vertical coefficients of permeability obtained from these

values are also described.

I. Introduction

Many natural and compacted cohesive soils are layered, and their consolidation behaviors are directionally dependent on the layering. 1)2)

In general, permeability may be several times less when the flow is perpendicu-lar rather than parallel to the layering. Therefore, it is said that the horizontal coefficient of consolidation, ch, is also many times greater than the vertical one,

cv. 3) For mesuring consolidation behaviors, it is unreasonable to adopt the conventional one-dimensional consolidation procedure. There is a need for an alternative method to obtain both values of ch and cv using of anisotropy. Several attempts have been made to obtain the ch value which will provide the sand-pile drainage design: (1) rotating the specimen 90-degrees, from its field orientation and testing in a normal oedometer, 4) (2) testing in a porous ring oedometer with solid platens, 5) (3) simulating both radial drainage and vertical loading by inser-tion of a small diameter drain of the sand column in the oedometer specimen. 6) In the case of rotation of the specimen, as the tests are all performed on an individual specimen, the differences in permeabilities are attributable not only to anisotropy

*Faculty of Bioresources , Mie University(Manuscript Received June 8, 1990, Accepted January 7, 1991)

Irrigation Engineering and Rural Planning No.21, 1991

AN EXAMPLE OF TWO-DIMENSIONAL CONSOLIDATION5

but also to testing error and statistical variations among the specimens. 2) The

use of a porous ring oedometer suffers from the serious problem that full side

friction develops during the test. Also, the presence of a vertical column of sand,

having different compressibilities compared with clay, seriouslly influences the

test results.

To obtain two measurements to determine the values of ch and cv using in-

dependent specimens, two-dimensional consolidation tests are carried out using

an oedometer ring which has the possibility of horizontal drainage as well as

vertical. In addition, to derive the consolidation settlement-time relationships

obtained from them, a numerical calculation method for determining the values of

ch and cv is described.

II. Soil Sample and Testing Procedure

Physical properties of the soil sample used for this study are given in Table 1.

The sample is prepared by sieving through a No. 4 sieve (the maximum size of soil

particles is 4.76mm). The consolidation tests were performed on two specimens

with a given water content. (1) The first is a specimen whose initial water

content w is the liquid limit and wet unit weight ƒÁt is 18.4kN/m3 (1.872g/cm3).

Hereafter, this sample will be referred to as the saturated sample. (2) The second

Table 1 Identification properties of the soil sample

Figure 1 Schematic drawing of lateral drainage type of oedometer ring


6 S. INOUE

is a specimen for which w is 24.0 %, ƒÁt is 19.3kN/m3 (1.968g/cm3), and degree of

saturation 94.4 %. This latter sample is referred to as the D90 sample. D90

means that the ratio of a given dry density to the maximum one on a compaction

curve of the soil sample is 0.90.

Each specimen is formed with a vibrating compactor in the oedometer ring in

five lifts of approximately equal thicknesses. A large oedometer whose ring is 15

cm in height and 30cm in diameter is used. 7) To simulate two-directional

drainage, the author adopted a lateral drainage type of ring with 24 longitudinal

slits spaced at 15-degree intervals around its periphery. The height of each slit is

10cm and its width about 1.5•`2.0mm. A schematic drawing of this ring is

shown in Figure 1. The inside surface of the ring is Teflon-coated and greased

prior to soil placement to minimize side friction during testing. There was no

leakage of soils even for the pasty sample since the inside portion of the slits was

coverd with filter paper. As a rule, the total testing procedure was based on the

method of the standard consolidation test according to the JIS established in

Japan, 1980.

III. Fundamental Equations

In the two-(or axisymmetrical three-) dimensional consolidation test for both the vertical and radial flow, it is assumed that the Terzaghi-Rendulic theory is valid.8) Thus, the differential equation govering the excess pore pressure u(r,z,t) in the specimens is given as follows:

(1)

where ch and cv are the r-directional (horizontal) and the z-directional (vertical) co-efficients of consolidation, respectively, t is the time coordinate, and r and z are the cylindrical coordinates as defined in Figure 2.

Figure 2 Boundary conditions of a consolidation specimen


AN EXAMPLE OF TWO-DIMENSIONAL CONSOLIDATION 7

In solving Eq. (1), Carrillo's method may be effective. 9) However, for conve-nience in carrying out the numerical calculation explained later, an alternative method is introduced. Assuming homogeneous anisotropy and taking the follow-ing transformation of variable z 10):

Equation (1) then becomes:

(2)

with boundary condition as shown in Figure 2:

u=0 at z=0 and at ξ=0

u=0 at z=2H and at ξ=2H√ch/cv

∂u/∂r=0 at r=0

u=0 at r=a

and initial condition:

u=uo when t=0

where H is one-half of the height of the specimen, a is the radius of the specimen and u0 is the initial pore pressure in the soil.

Under these boundary and initial conditions, the solution of Eq. (2) leads to the following expression for pore pressure distribution in the specimen:

(3)

where ƒÊn'S are roots of Jo(ƒÊn• a)=0(n=1,2,•c). J0 and J1 are symbols of the Bessel

function of the 1st kind at zero and the first orders, respectively,

M m=(2m-1)/2π,(m=1,2,…)

and λmn2=v2m+(μn/a)2, vm=(2m-1)・ π√cv/ch/2H

Thus, the average degree of consolidation U due to two-dimensional flow is

derived as follows:

(4)

where ch•Et/a2=Th, Cv• t/H2=Tv.

Putting ƒÉv=cv/H2 and ƒÉh=ch/a2, Eq. (4) may be rearranged to give:

(5)

On the other hand, the pore pressure distribution derived from the classical

theory of one-dimensional consolidation, as is generally known, is given by:


8 S. INOUE

(6)

and the corresponding average degree of consolidation is as follows:

(7)

The letter symbols used in Eq. (6) and (7) are the same where they first appear.

IV. A Numerical Calculation Method to Determine the Coefficient of Consolidation

Based on Eqs. (5) snd (7), the settlement and time relations obtained from one-and two-dimensional consolidation tests are analyzed to determine the ch and cv values.

First of all, the cv value is determined from the one-dimensional consolidation test. In this case, it is more convenient to carry use a numerical calculation method based on least squares than a graphic or chart method, such as the curve rule method.u) Its advantage is that error due to judgment can be reduced and more accurate numerical results can be obtained in comparison with the latter method.

A method based on numerical calculation has been developed by the author for

the determination of the cv values of one-dimensional consolidation testing

results. 12)

For two-dimensional problems the above described method is applied as follows. If it is assumed that the A, value corresponding to cv is already determined, the

first approximation of the All value may be derived from the following equation as a product of two terms up to the the second order of the infinite series of the right-hand side of Eq. (5):

(8)

in which d is the oedometer dial gauge reading of sample compression, A and B are

constant, and M1 and M2 are values corresponding to m=1 and 2.

To develop this equation further, the following expressions are defined by

means of substitution:

(9)

Now, taking time t in terms of an arithmetical progression to common difference

ƒÑ:

t=to+(i-1)τ

where i=1, 2, 3,•c, and to is a initial value of t.

And denoting d1, d2,•c, di-1, di and di+1 for measured values of the consolidation

settlement at each interval of time, then Eq. (8) leads to the following:



(10)

Let yi+1/2, yi+3/2,•cyi+9/2 be defined by the expression yi+1/2=di+i-di,

yi+3/2=di+2-di+1,•c, yi+9/2=di+5-di+4 and putting:

(11)

Eq. (10) then may be rearranged to give:

(12)

where g1=-B•E[f1ƒÉv•E{to+(i-1)•EƒÑ}•Ef3ƒÉh•E{to+(i-1)•EƒÑ}•E(f1ƒÉv-ƒÑ•Ef3ƒÉh• ƒÑ-1)], etc.

By elimination of gi g2, g3 and g4 in Eq. (12), the next relation leads to:

(13)

in which

(14)

These a, ƒÀ, ƒÁ and 6 become the coefficient of the following equation of the fourth

degree equation regarding variable z:

(15)

Therefore, zi, z2, z3 and z4 are roots of this equation. Producing a normal

equation of Eq. (13), the values of ƒ¿, ƒÀ, ƒÁ and ƒÂ can be uniquely determined. And

by the solution of the Eq. (15), the first approximations of the value of )h can be

obtained from Eq. (11) if the value of is known. To incorporate all the above-

mentioned requirements nine more date points are necessary.

As the following relation is derived from the first equation of Eq. (11),

ƒÉh =-1

/τ ・μ21{ln(z1)+λv・ τ・M21}

four kinds of the first approximation value of th can be determined from the other

three relations in the same manner. To select a suitable and more accurate value

for .h, one changes such obtained values in the vicinity of the first approximation


10S. INOUE

vlaue and finds a unique vlaue through an iterative calculation. The approach

which may be used to achive these requirements is to make the total sum of the

squares of the residual differences between the claculated values and measured

data as small as possible. Consequently, both coefficients ch and cv, best fitted to

the settlement versus time relationships can be readily computed from the

relations of ch=a2•EƒÉh and cv,=H2•EƒÉ

vFinally, form Eq. (8), when t=0, a corrected initial reading do is do=A-B and

when t•‡, a final reading of primary consolidation d100=A.

V. Results and Discussion

As examples of the use of the above-mentioned method, the typical relationships between the consolidation settlement and time are shown in Figure 3 in the case of the consolidation pressure p=39.2 kN/m2 (0.40 kgf/cm2) and in Figure 4 in that of 78.4 kN/m2 (0.80 kgf/cm2), compared with the measured and calculated

Figure 3 Relationship between settlement and time in the case of p=39.2 kN/m2

Figure 4 Relationship between settlement and time in the case of p=78.5 kN/m2



values for both one-and two-dimensional cases. From these figures, it can be

seen that the calculated curves fit very well to the measured values in the region

of the primary consolidaton for both one-and two-dimensional consolidation.

As a consequence of the examination of consolidation behavior of both the

saturated and D90 specimens for each level of consolidation pressure from 9.8 kN/

m2 (0.1 kgf/cm2) to 1255 kN/m2 (12.8 kgf/cm2), the amount of the settlement due to

two-dimensional consolidation is, on the whole, greater than that due to one-

dimensional consolidation. It is considered that this phenomenon may be greatly

affected by the amount of primary consolidation •¢d' (=d100-do). Thus, the values

of •¢d' are evaluated as presented in Table 2 for both specimens.

Table 2 Comparison of the amount of the primary consolidation •¢d'

Figure 5 ch, cu and mu versus log of mean consolidation pressure relationship

Irrigation Engineering and Rural Planning No. 21, 1991

12 S. INOUE

The table indicates that the •¢d' values of the two-dimensional consolidation are

generally larger than those of the one-dimensional one: this implies that the

progress of the two-dimensional consolidation is more rapid. Values of ch and cu

on two specimens for mean consolidation pressure fi, and values of the coefficient

of volume compressibility mu common to two specimens are shown in Figure 5.

However, values of mu contain the secondary compression. It can be readily seen

that the cu values are nearly constant for the P for the saturated and D90

specimens.

In contrast, the difference between the ch vlaues of the two samples tends to

reduce gradually as p increases. This tendency may indicate that anisotropy of

the D90 specimen is more remarkable than that of the saturated specimen since the D90 specimen was compacted with layering when the oedometer ring was

filled with soil. In the saturated specimen there is no compaction effect because

of the pasty sample. The anisotropic behavior tends to be reduced gradually as

p increases. The relation between the ratio ch/cu and p is plotted in Figure 6 to

Figure 6 Correlation of ch/cu ratio and log of mean consolidation pressure

Figure 7 kh and ku versus log of mean consolidation pressure relationship



estimate the differences in the vlaues of ch and cu.

For two samples, the ratio is generally in the range between ch/cu=15 and 50,

with the average of about 30, though there is a slight scatter.

Finally, variations of the vertical and horizontal permeability coefficient ku and

kh, respectively, computed by the following two equations are shown in Figure 7

for p:

ku=ƒÁƒÖ•EmƒÒ•EcƒÒ, kh=ƒÁƒÖ•EMƒÒ•ECh

where it is assumed that the value of mƒÒ is common to both directions under the

confined compression condition. ƒÁƒÖ is the unit weight of water.

As the values of mƒÒ in the above-mentioned equations, test data plotted in

Figure 5 are used. There is no significant difference in mƒÒ values between the

saturated and the D90 sample. The values of kh are about 8 to 17 times kƒÒ.

VI. Conclusions

In an attempt to determine the vertical and horizontal coefficients of consolidation

cƒÒ and ch, ues of the lateral drainage type of oedometer ring is proposed. A

numerical calculation method is also proposed by fitting an equaton to estimate

the degree of consolidation derived from the two-(or axisymmetrical three-)

dimensional consolidation theory to the relationships between the consolidation

settlement and time obtained from this test. For the saturated sample and the

unsatuated D90 one, the following conclusions can be drawn:

(1) The horizontal coefficient of consolidation, ch, can be readily calculated by

the numerical calculation method for the relationship between two-dimensional

consolidation settlement and time provided the vertical coefficient, cƒÒ, for the one-

dimensional problem has been obtained.

(2) An examination of the cƒÒ and ch values for both the saturated and the D90

samples reveals that the D90 sample with layering compaction is more anisotropic

than the saturated sample.

(3) Based on the test results, it is concluded that the values of ch are about 15 to

50 times that of cƒÒ and that the horizontal coefficients of permeability, kh are 8 to

17 times the vertical ones, kƒÒ, for each sample.

Further one-and two-dimensional consolidation tests need to be performed on

the same soil sample. Based on this reasoning, it is thought that the differences

in measurements can be attributed not only to anisotropy but also to testing error

among the specimens. To solve such problems, there is a need further improve-

ment in future work.

References

1) Rowe, P. W.: The calculation of the consolidation reats of laminatd, varved or layered clays, with particular reference to drains, Geotechnique Vol. 14, No. 4, pp. 321-340

(1964)2) Peters, J. F. and Leavell, D. A.: A biaxial consolidation test for anisotropic soils, ASTM


14 S. INOUE

special technical publication; 892, pp. 465-484 (1986)

3) Wu, T. H., Chang, N. Y. and Ali, A. M.: Consolidation and strength properties of clays, Proc., ASCE, Vol. 104, No. GT7, pp. 889-905 (1978)

4) Rowe, P. W.: Measure of the coefficient of consolidation of lacustrine clay, Geotech-nique, Vol. 9, No. 3, pp. 107-118 (1959)

5) McKinley, D. G.: A laboratory study of rates of consolidation in clays with particular reference to conditions of radial pore water drainage, Proc. 5th Int. Conf. Soil Mech.,

Vol. 1, pp. 225-228 (1961)6) Shields, D. H. and Rowe, P. W.: Radial drainage oedomater for laminated clays, Proc.

ASCE, Vol. 91, No. SM1, pp. 15-23 (1965)7) Inoue, S.: Consolidation behaviors of compacted soils using a large scale oedometer,

Bull. Fac. Bioresources, Mie Univ., No. 1, pp. 51-62 (1988)8) Sills, G. C.: Some conditions under which Biot's equatins of consolidation reduce to

Terzaghi's equation, Geotechnique, Vol. 25, No. 1, pp. 129-132 (1975)9) Guido, V. A. and Ludewig, N. M.: A comparative labolatory evaluation of band-shaped

prefabricated drains, ASTM special technical publication; 892, pp. 642-669 (1986)10) Wilkinson, W. B.: Constant head in situ permeability tests in clay strata, Geotech-

nia ue, Vol. 18, No. 2, pp. 172-194 (1968)11) Mikasa, M. and Takada, N.: Determination of coefficient of consolidation (cƒÒ) for large

strain and variable cƒÒ values, ASTM special techical publication; 892, pp.526-547

(1986)

12) Inoue, S.: Evaluation of the consolidated defromation and time relations using a method of numerical calculation, Tsuchi-to-Kiso, JSSMFE, Vol.34, No.12, pp.57-61

(1984) (in Japanese)


an example of two-dimensional consolidation testing and

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